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Chapter 9: Problem 43

Solve equation by using the square root property. Simplify all radicals. \((3 x+2)^{2}=49\)

Short Answer

Expert verified
The solutions are \(x = \frac{5}{3}\) and \(x = -3\).

Step by step solution

01

Rewrite the equation

Start with the given equation \((3x+2)^2 = 49\)
02

Apply the square root property

Take the square root of both sides to get \[\sqrt{(3x+2)^2} = \pm\sqrt{49}\]
03

Simplify the radicals

\[|3x+2| = 7\ \Rightarrow \ 3x + 2 = 7 \text{ or }\ 3x + 2 = -7\]
04

Solve for x

Solve the first equation: \[3x + 2 = 7 \ \Rightarrow\ 3x = 5 \ \Rightarrow\ x = \frac{5}{3}\]Solve the second equation:\[3x + 2 = -7 \ \Rightarrow\ 3x = -9 \ \Rightarrow\ x = -3\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving quadratic equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). In this exercise, we have a slightly different form: \((3x+2)^2 = 49\). To solve these, we can use the square root property, which states that if \(u^2 = k\), then \(u = \pm \sqrt{k}\). This allows us to turn our quadratic equation into a simpler linear one. By taking the square root of both sides, we get two potential solutions to check. Remember, the square root of a number has both a positive and a negative value.
simplifying radicals
Simplifying radicals means expressing the radical in its simplest form. Here, we encounter the radical \(\sqrt{49}\). Since 49 is a perfect square, its square root is 7. So, we simplify \(\sqrt{49}\) to 7. Another essential concept is applying the absolute value, i.e., \(\sqrt{(3x+2)^2} = |3x+2|\). This helps to resolve the equation into two distinct possibilities: \(3x+2 = 7\) and \(3x+2 = -7\).
absolute value equations
Absolute value equations involve expressions within absolute value signs. In the step \(|3x+2| = 7\), we interpret this as two separate equations: \(3x+2 = 7\) or \(3x+2 = -7\). This approach stems from the property that the absolute value of a number is always non-negative and represents the distance from zero on the number line. Solving these two equations separately gives us the potential solutions for x.
steps to solve equations
To solve the equation \((3x+2)^2=49\) using the square root property, follow these steps:

1. Start by rewriting the given equation.
2. Apply the square root property to both sides of the equation.
3. Simplify the resulting radicals.
4. Solve the absolute value equation by separating it into two simpler linear equations.
5. Finally, solve each of these linear equations for x.

In our example:
Step 1: \((3x+2)^2=49\)
Step 2: \(\sqrt{(3x+2)^2}=\pm\sqrt{49}\)
Step 3: \(|3x+2|=7\)
Step 4: Solve \(3x+2=7\) and \(3x+2=-7\)
Step 5: The solutions are \(x=\frac{5}{3}\) and \(x=-3\).

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. (a) Give solutions in exact form, and (b) use a calculator to give solutions correct to the nearest thousandth. $$ x^{2}=2+4 x $$

Use a calculator with a square root key to solve each equation. Round your answers to the nearest hundredth. \((2.11 p+3.42)^{2}=9.58\)

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ 5 x^{2}+3=2 x $$

The amount A that \(P\) dollars invested at an annual rate of interest r will grow to in 2 yr is \(A=P(1+r)^{2}\). At what interest rate will \(\$ 500\) grow to \(\$ 530.45\) in 2 yr?

Solve equation by using the square root property. Simplify all radicals. \((3 k+1)^{2}=18\)
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